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E = mc2 Explained

Albert Einstein
is perhaps the most famous scientist of this century. One of his most well-known accomplishments is the formula
Despite its familiarity, many people don't really understand what it means. We hope this explanation will help!
One of Einstein's great insights was to realize that matter and energy are really different forms of the same thing. Matter can be turned into energy, and energy into matter.
For example, consider a simple hydrogen atom, basically composed of a single proton. This subatomic particle has a mass of
0.000 000 000 000 000 000 000 000 001 672 kg
This is a tiny mass indeed. But in everyday quantities of matter there are a lot of atoms! For instance, in one kilogram of pure water, the mass of hydrogen atoms amounts to just slightly more than 111 grams, or 0.111 kg.
Einstein's formula tells us the amount of energy this mass would be equivalent to, if it were all suddenly turned into energy. It says that to find the energy, you multiply the mass by the square of the speed of light, this number being 300,000,000 meters per second (a very large number):

= 0.111 x 300,000,000 x 300,000,000
= 10,000,000,000,000,000 Joules
This is an incredible amount of energy! A Joule is not a large unit of energy ... one Joule is about the energy released when you drop a textbook to the floor. But the amount of energy in 30 grams of hydrogen atoms is equivalent to burning hundreds of thousands of gallons of gasoline!

If you consider all the energy in the full kilogram of water, which also contains oxygen atoms, the total energy equivalent is close to 10 million gallons of gasoline!
Can all this energy really be released? Has it ever been?

The only way for ALL this energy to be released is for the kilogram of water to be totally annhilated. This process involves the complete destruction of matter, and occurs only when that matter meets an equal amount of antimatter ... a substance composed of mass with a negative charge. Antimatter does exist; it is observable as single subatomic particles in radioactive decay, and has been created in the laboratory. But it is rather short-lived (!), since it annihilates itself and an equal quantity of ordinary matter as soon as it encounters anything. For this reason, it has not yet been made in measurable quantities, so our kilogram of water can't be turned into energy by mixing it with 'antiwater'. At least, not yet.

Another phenomenon peculiar to small elementary particles like protons is that they combine. A single proton forms the nucleus of a hydrogen atom. Two protons are found in the nucleus of a helium atom. This is how the elements are formed ... all the way up to the heaviest naturally occuring substance, uranium, which has 92 protons in its nucleus.
It is possible to make two free protons (Hydrogen nuclei) come together to make the beginnings of a helium nucleus. This requires that the protons be hurled at each other at a very high speed. This process occurs in the sun, but can also be replicated on earth with lasers, magnets, or in the center of an atomic bomb. The process is called nuclear fusion.
What makes it interesting is that when the two protons are forced to combine, they don't need as much of their energy (or mass). Two protons stuck together have less mass than two single separate protons!
When the protons are forced together, this extra mass is released ... as energy! Typically this amounts to about 0.7% of the total mass, converted to an amount of energy predictable using the formula .

Elements heavier than iron are unstable. Some of them are very unstable! This means that their nuclei, composed of many positively charged protons, which want to repel from each other, are liable to fall apart at any moment! We call atoms like this radioactive.
Uranium, for example, is radioactive. Every second, many of the atoms in a chunk of uranium are falling apart. When this happens, the pieces, which are now new elements (with fewer protons) are LESS massive in total than the original uranium atoms. The extra mass disappears as energy ... again according to the formula ! This process is called nuclear fission.

Both these nuclear reactions release a small portion of the mass involved as energy. Large amounts of energy! You are probably more familiar with their uses. Nuclear fusion is what powers a modern nuclear warhead. Nuclear fission (less powerful) is what happens in an atomic bomb (like the ones used against Japan in WWII), or in a nuclear power plant.

Albert Einstein was able to see where an understanding of this formula would lead. Although peaceful by nature and politics, he helped write a letter to the President of the United States, urging him to fund research into the development of an atomic bomb ... before the Nazis or Japan developed their own first. The result was the Manhatten Project, which did in fact produce the first tangible evidence of ... the atomic bomb!

Hear Albert Einstein state his famous theorem: download an mp3 (104k, zipped) here.

See also: The Birth of Atoms | Einstein's Theory of Relativity

Physics | Science Page | Worsley School



"If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world. Should my theory prove untrue, France will say that I am a German and Germany will declare that I am a Jew." - Albert Einstein ( 1879 - 1955 )

"It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing -- a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned above. This was demonstrated by Cockcroft and Walton in 1932, experimentally." - Prof. Albert Einstein ( excerpts from 1947 film, "Atomic Physics" )


source: American Institute of Physics ( )

Special Relativity was first published in 1905 by Albert Einstein at age 26 working quietly in the Swiss Patent Office, Bern, Switzerland, under the title "On The Electrodynamics Of Moving Bodies", translated from "Zur Elektrodynamik bewegter Körper", Annalen der Physik, volume 17: 891, 1905, a downloadable copy of which is available here in pdf.

Also read "On The Relativity Principle And The Conclusions Drawn From It", by A. Einstein, translated from Jahrbuch der Radioaktivität und Elektronik volume 4 (1907): 411-462

And, "Does the Inertia of a Body Depend upon its Energy-Content?", by A. Einstein, Annalen der Physik volume 18: 639, 1905

§ Define:


§ Some Derivations:


derivation 1.png.


derivation 2.png

§ The Problem:

However the entire classical Newtonian physics derived above is predicated upon the concept of mass as an invariant constant. But we now know differently, namely that mass, m, is a variable quantity owing to the Addition of Relativistic Velocities, where 

variable mass.png

is the relationship between rest mass undergoing velocity and its equivalent dilated mass. 

§ The Solution:

derivation 3.png

But, whoa! Look,


§ More Simple Algebraic Derivation:

note: see another quick and dirty matheamtical derivation

binomial expansion series

§ Einstein's Interpretation:

The interpretation that Einstein therefore applied is as follows: 

reinterpretation of energy

Nevertheless it still should always be remembered that 


On the other hand, applying a relativistic kinetic energy concept, we can arrive at the following: 

relativistic k.e.

§ The Law of Inertia of Energy:

Law of Energy Inertia


inertial mass

is the equation for matter in the form of inertial ( dilated ) mass which can be derived from a given amount of energy E  whose capability for performing work is given by



§ Derivation of classical Newtonian kinetic energy:

classical kinetic energy

§ Derivation of relativistic energy:

However for , relativistic mass dilation as a function of velocity,

where is rest mass ( proper mass ) within a given inertial frame of reference, we still have

However using a dummy variable trick for integrating,

integral calculus

And therefore,

relativistic kinetic energy

§ 2nd Derivation of relativistic energy:

§ Derivation of classical kinetic energy:

mathematical references

§ Here we now have these important energy definitions:


§ Derive the law of conservation of total energy, relativistic and non - relativistic:

Finally, using the binomial series to derive the law of conservation of total energy, relativistic and non - relativistic:


§ Deriving mass dilation using Richard Fehnman's suggested equations from his "Lectures on Physics - Vol. I " ( although this derivation is somewhat recursive ):


"Imagination is more important than knowledge" - Albert Einstein ( 1879 - 1955 )


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